For \(u\in \Sigma _a^{+,*}\)  and a set \(B\in \mathcal {B}_a^+\)  we denote by
\(uB=\lbrace u\mathbf {i}^+:\mathbf {i}^+\in B\rbrace .\) 
By using [1]} we have that for \(\mathbf {i}^+==i_1i_2\cdots \in \Sigma _a^+\)  and \(r>0\) ,
\(\Phi _{\mathbf {I}}^{-1}(B(\Phi _{\mathbf {I}}(\mathbf {i}),r_{\mathbf {i}^+|_1}r))\cap \mathcal {P}_a^+(\mathbf {i}^+)=&\ \sigma ^{-1}\Phi _{\mathbf {I}}^{-1}(B(\Phi _{\mathbf {I}}(\sigma \mathbf {i}^+),r))\cap \mathcal {P}_a^+(\mathbf {i}^+)\nonumber \\=&\ i_1\Phi _{\mathbf {I}}^{-1}(B(\Phi _{\mathbf {I}}(\sigma \mathbf {i}^+),r)).\) 

For \(n\ge 0\)  we shortly denote by
\(B_{{\mathbf {I}}}(\mathbf {i}^+,n)=\Phi _{\mathbf {I}}^{-1}(B(\Phi _{\mathbf {I}}(\mathbf {i}^+),R_{\mathbf {I}}\cdot r_{\mathbf {i}^+|_n})),\) 
where \(r_{\mathbf {i}^+|_n}=r_{i_1}\cdots r_{i_n}\)  for \(n\ge 1\)  and \(r_{\emptyset }=1\) . Then by (REF ) we have that for \(n\ge 1\)  and \(0\le k\le n-1\) ,
\(B_{{\mathbf {I}}}(\sigma ^k\mathbf {i}^+,n-k)\cap \mathcal {P}_a^+(\sigma ^k\mathbf {i}^+)=&\ \sigma ^{-1} B_{{\mathbf {I}}}(\sigma ^{k+1}\mathbf {i}^+,n-k-1)\cap \mathcal {P}_a^+(\sigma ^k\mathbf {i}^+)\\=&\ i_{k+1}B_{{\mathbf {I}}}(\sigma ^{k+1}\mathbf {i}^+,n-k-1).\) 

This implies
\(\frac{\bar{\mu }_{\mathbf {i}^-,\mathbf {\omega }}^+(\mathbf {i}^+|_kB_{{\mathbf {I}}}(\sigma ^k\mathbf {i}^+,n-k))}{\bar{\mu }_{\mathbf {i}^-,\mathbf {\omega }}^+(\mathbf {i}^+|_{k+1}B_{{\mathbf {I}}}(\sigma ^{k+1}\mathbf {i},n-k-1))}=&\ \frac{\bar{\mu }_{\mathbf {i}^-,\mathbf {\omega }}^+(\mathbf {i}^+|_kB_{{\mathbf {I}}}(\sigma ^k\mathbf {i}^+,n-k))}{\bar{\mu }_{\mathbf {i}^-,\mathbf {\omega }}^+(\mathbf {i}^+|_k(B_{{\mathbf {I}}}(\sigma ^k\mathbf {i},n-k)\cap \mathcal {P}_a^+(\sigma ^k\mathbf {i}^+)))}.\) 

For \(u, v\in \Sigma _a^{+,*}\)  with \(\widetilde{\mu }^+_{\mathbf {i}^-,\mathbf {\omega }}([u])>0\) , by (REF ) we have
\(\bar{\mu }_{\mathbf {i}^-,\mathbf {\omega }}^+([uv])=\frac{\widetilde{\mu }^+_{\mathbf {i}^-,\mathbf {\omega }}([uv])}{\widetilde{\mu }^+_{\mathbf {i}^-,\mathbf {\omega }}([u])}=\frac{Q_u(\mathbf {\omega })\cdot \mu ^+_{\mathbf {i}^-}([u])\cdot \widetilde{\mu }_{\mathbf {i}^-u,\mathbf {\omega }_u}^{+}([v])}{Q_u(\mathbf {\omega })\cdot \mu ^+_{\mathbf {i}^-}([u])\cdot \widetilde{\mu }_{\mathbf {i}^-u,\mathbf {\omega }_u}^{+}(\Sigma _a^+)}=\bar{\mu }_{\mathbf {i}^-u,\mathbf {\omega }_u}^{+}([v]).\) 
This implies that for \(\mu ^{-}\) -a.e. \(\mathbf {i}^-\)  and \(\mathbb {P}_a\) -a.e. \(\mathbf {\omega }\) , given \(\widetilde{\mu }^+_{\mathbf {i}^-,\mathbf {\omega }}([\mathbf {i}|_n])>0\)  we have
\(\frac{\widetilde{\mu }^+_{\mathbf {i}^-,\mathbf {\omega }}(B_{{\mathbf {I}}}(\mathbf {i},n))}{\widetilde{\mu }^+_{\mathbf {i}^-,\mathbf {\omega }}([\mathbf {i}|_n])}=&\ \prod _{k=0}^{n-1} \frac{\bar{\mu }^+_{\mathbf {i}^-,\mathbf {\omega }}(\mathbf {i}^+|_kB_{{\mathbf {I}}}(\sigma ^k\mathbf {i}^+,n-k))}{\bar{\mu }^+_{\mathbf {i}^-,\mathbf {\omega }}(\mathbf {i}^+|_{k+1}B_{{\mathbf {I}}}(\sigma ^{k+1}\mathbf {i}^+,n-k-1))}\nonumber \\=&\ \prod _{k=0}^{n-1}\frac{\bar{\mu }^+_{\mathbf {i}^-,\mathbf {\omega }}(\mathbf {i}^+|_kB_{{\mathbf {I}}}(\sigma ^k\mathbf {i}^+,n-k))}{\bar{\mu }^+_{\mathbf {i}^-,\mathbf {\omega }}(\mathbf {i}^+|_k(B_{{\mathbf {I}}}(\sigma ^k\mathbf {i}^+,n-k)\cap \mathcal {P}^+(\sigma ^k\mathbf {i}^+)))}\nonumber \\=&\ \prod _{k=0}^{n-1}\frac{\bar{\mu }_{\mathbf {i}^-i_1\cdots i_k,\mathbf {\omega }_{i_1\cdots i_k}}^{+}(B_{{\mathbf {I}}}(\sigma ^k\mathbf {i}^+,n-k))}{\bar{\mu }_{\mathbf {i}^-i_1\cdots i_k,\mathbf {\omega }_{i_1\cdots i_k}}^{+}(B_{{\mathbf {I}}}(\sigma ^k\mathbf {i}^+,n-k)\cap \mathcal {P}_a^+(\sigma ^k\mathbf {i}^+))}.\) 

For \(n\ge 0\)  define the non-negative measurable function on \(\Sigma _a\times \Omega _a\) :
\(f_n(\mathbf {i},\mathbf {\omega })=-\log \frac{\bar{\mu }^+_{\mathbf {i}^-,\mathbf {\omega }}(B_{{\mathbf {I}}}(\mathbf {i}^+,n)\cap \mathcal {P}_a^+(\mathbf {i}^+))}{\bar{\mu }^+_{\mathbf {i}^-,\mathbf {\omega }}(B_{{\mathbf {I}}}(\mathbf {i}^+,n))},\) 
with the convention that \(\frac{0}{0}=1\) . For \(k\ge 1\)  we have for \((\mathbf {i},\mathbf {\omega })\in \Sigma _a\times \Omega _a\) ,
\(\widetilde{\sigma }^k(\mathbf {i},\mathbf {\omega })=(\sigma ^k\mathbf {i}, \mathbf {\omega }_{i_1\cdots i_k}).\) 
Therefore
\(f_{n-k}\circ \widetilde{\sigma }^k(\mathbf {i},\mathbf {\omega })=\log \frac{\bar{\mu }_{\mathbf {i}^-i_1\cdots i_k,\mathbf {\omega }_{i_1\cdots i_k}}^{+}(B_{{\mathbf {I}}}(\sigma ^k\mathbf {i}^+,n-k))}{\bar{\mu }_{\mathbf {i}^-i_1\cdots i_k,\mathbf {\omega }_{i_1\cdots i_k}}^{+}(B_{{\mathbf {I}}}(\sigma ^k\mathbf {i}^+,n-k)\cap \mathcal {P}_a^+(\sigma ^k\mathbf {i}^+))}.\) 
Thus by (REF ) we obtain that
\(\log \frac{\widetilde{\mu }^+_{\mathbf {i}^-,\mathbf {\omega }}(B_{{\mathbf {I}}}(\mathbf {i},n))}{\widetilde{\mu }^+_{\mathbf {i}^-,\mathbf {\omega }}([\mathbf {i}|_n])} = \frac{1}{n}\sum _{k=0}^{n-1} f_{n-k}\circ \widetilde{\sigma }^k(\mathbf {i},\mathbf {\omega }).\) 
By Proposition REF  we have that for \(\mu ^{-}\) -a.e. \(\mathbf {i}^-\) , for \(\mathbb {P}_a\) -a.e. \(\mathbf {\omega }\)  with \(\Vert \widetilde{\mu }_{\mathbf {i}^-,\mathbf {\omega }}\Vert >0\) , for \(\bar{\mu }_{\mathbf {i}^-,\mathbf {\omega }}\) -a.e. \(\mathbf {i}^+\) ,
\(\lim _{n\rightarrow \infty }f_n(\mathbf {i}^-,\mathbf {i}^+,\mathbf {\omega })=I_{\bar{\mu }^+_{\mathbf {i}^-,\mathbf {\omega }}}(\mathcal {P}_a^+|\mathcal {B}_{\mathbf {I}})(\mathbf {i}^-,\mathbf {i}^+),\) 
and there is a \(L^1(\Sigma _a^+,\mathcal {B}_a^+,\bar{\mu }^+_{\mathbf {i}^-,\mathbf {\omega }})\)  function \(g=g(\mathbf {i}^-,\cdot ,\mathbf {\omega })\)  with \(\int g \,\mathrm {d}\bar{\mu }^+_{\mathbf {i}^-,\mathbf {\omega }} \le H_{\bar{\mu }^+_{\mathbf {i}^-,\mathbf {\omega }}}(\mathcal {P}_a^+)+C\le a\log a+C\)  such that \(f_n\le g\)  holds for all \(n\ge 0\) . Hence by the exact-dimensionality of \(\mu ^-\) -a.e. \(\widetilde{\mu }^+_{\mathbf {i}^-,\mathbf {\omega }}\)  on \((\Sigma _a^+,d_\delta ^+)\)  from Theorem REF , and then by Lemma REF  and Theorem REF , we have that for \(\mathbb {Q}_\mu \) -a.e. \((\mathbf {i},\mathbf {\omega })\) , or equivalently, for \(\mu ^-\) -a.e. \(\mathbf {i}^-\) , for \(\mathbb {P}_a\) -a.e. \(\mathbf {\omega }\)  with \(\Vert \widetilde{\mu }^+_{\mathbf {i}^-,\mathbf {\omega }}\Vert >0\) , for \(\bar{\mu }^+_{\mathbf {i}^-,\mathbf {\omega }}\) -a.e. \(\mathbf {i}^+\) ,
\(\lim _{n\rightarrow \infty }\frac{1}{n}\log \widetilde{\mu }_{\mathbf {i}^-}(B_{{\mathbf {I}}}(\mathbf {i}^+,n))=&\ \lim _{n\rightarrow \infty }\frac{1}{n}\log \widetilde{\mu }^+_{\mathbf {i}^-}([\mathbf {i}^+|_n])+\frac{1}{n}\sum _{k=0}^{n-1} f_{n-k}\circ \widetilde{\sigma }_a^k(\mathbf {i}^-,\mathbf {i}^+,\mathbf {\omega })\\=&\ -(h_{\mu }-h_V)+h_{\mu ,V,\mathbf {I}}.\) 

Finally, for the Lyapunov exponent, we have that, by Birkhoff ergodic theorem and (REF ), \(\mathbb {Q}_\mu \) -a.s.,
\(\lim _{n\rightarrow \infty }\frac{-\log r_{\mathbf {i}|_n}}{n}=&\ \int _{\Omega _a}\int _{\Sigma _a^-}\int _{\Sigma _a^+} -\log r_{\mathbf {i}^+|_1} \, \widetilde{\mu }^+_{\mathbf {i}^-,\mathbf {\omega }}(\mathrm {d}\mathbf {i}^+)\mu ^-(\mathrm {d}\mathbf {i}^-)\mathbb {P}_a(\mathrm {d}\mathbf {\omega })\\=&\ \int _{\Sigma _a^+} -\log r_{\mathbf {i}^+|_1} \, \mu ^+(\mathrm {d}\mathbf {i}^+)\\=&\ \sum _{i=1}^a -\mu ^+([i])\log r_i=\chi _{\mu ,\mathbf {I}}.\) 

This yields the conclusion.
